ME451 Kinematics and Dynamics

of Machine Systems

(Gears)

Cam-Followers and Point-Follower 3.4.1, 3.4.2

September 27, 2013

Radu SerbanUniversity of Wisconsin-Madison

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Before we get started…

Last time: Relative constraints (revolute, translational) Composite joints (revolute-revolute, revolute-translational)

Today: Gears Cam – Followers Point – Follower

Assignments: HW 5 – due September 30, in class (12:00pm) Matlab 3 – due October 2, Learn@UW (11:59pm)

Gears (convex-convex, concave-convex, rack and pinion)

3.4.1

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Gears Convex-convex gears

Gear teeth on the periphery of the gears cause the pitch circles shown to roll relative to each other, without slip

First Goal: find the angle , that is, the angle of the carrier

What’s known: Angles i and j

The radii Ri and Rj

You need to express as a function of these four quantities plus the orientation angles i and j

Kinematically: PiPj should always be perpendicular to the contact plane

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Gears - Discussion of Figure 3.4.2 (Geometry of gear set)

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Gears - Discussion of Figure 3.4.2 (Geometry of gear set)

Note: there are a couple of mistakes in the book, see Errata slide

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Gear Set Constraints

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Example: 3.4.1

Gear 1 is fixed to ground Given to you: 1 = 0 , 1 = /6, 2=7/6 , R1 = 1, R2 = 2

Find 2 as gear 2 falls to the position shown (carrier line P1P2 becomes vertical)

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Gears (Convex-Concave)

Convex-concave gears – we are not going to look into this class of gears

The approach is the same, that is, expressing the angle that allows on to find the angle of the

Next, a perpendicularity condition using u and PiPj is imposed (just like for convex-convex gears)

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Rack and Pinion Preamble

Framework: Two points Pi and Qi on body i

define the rack center line Radius of pitch circle for pinion is Rj

There is no relative sliding between pitch circle and rack center line

Qi and Qj are the points where the rack and pinion were in contact at time t=0

NOTE: A rack-and-pinion type kinematic

constraint is a limit case of a pair of convex-convex gears Take the radius Ri to infinity, and

the pitch line for gear i will become the rack center line

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Rack and Pinion Kinematics

Kinematic constraints that define the relative motion: At any time, the distance between

the point Pj and the contact point D should stay constant (this is equal to the radius of the gear Rj)

The length of the segment QiD and the length of the arc QjD should be equal (no slip condition)

Rack-and-pinion removes two DOFs of the relative motion between these two bodies

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Rack and Pinion Constraints

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Errata:

Page 73 (transpose and signs)

Page 73 (perpendicular sign, both equations)

Cam – Followers3.4.2

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Cam – Follower Pair

Setup: Two shapes (one on each body) that are always in contact (no chattering) Contact surfaces are convex shapes (or one is flat) Sliding is permitted (unlike the case of gear sets)

Modeling basic idea: The two bodies share a common point The tangents to their boundaries are collinear

Source: Wikipedia.org

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Interlude: Boundary of a Convex Shape (1)

Convex shape assumption any point on the boundary is defined by a unique value of the angle .

The distance from the reference point to any point on the convex boundary is a function of :

We need to express two quantities as functions of : The position of , that is The tangent at , that is

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In the LRF:

where

and therefore

[handout]

Interlude Boundary of a Convex Shape (2)

In the GRF:

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Cam – Follower Pair

Step 1 The two bodies share the contact point: (2 constraints)

The two tangents are collinear: (1 constraint)

Recall that points and are located by the angles i and j, respectively.

Therefore, in addition to the coordinates for each body, one needs to include one additional generalized coordinate, namely the angle :

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Cam – Follower Constraints

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Example 3.4.3

Determine the expression of the tangents g1 and g2

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Cam – Flat-Faced Follower Pair

A particular case of the general cam-follower pair Cam stays just like before Flat follower Typical application: internal combustion engine Not covered in detail, HW touches on this case

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Errata:

Page 83(Q instead of P)

Page 80(subscript ‘j’ instead of ‘i’)

Point – Follower3.4.3

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Point – Follower Pair

Setup: Pin , attached to body can move (slide

and rotate) in a slot attached to body Modeling basic idea:

Very similar to a revolute joint, except… …point moves on body Location of point on body is

parameterized by the angle Therefore, in addition to the coordinates

for body , one needs to include one additional generalized coordinate, namely the angle :

Note: this diagram is more general than theone in the textbook (includes point )

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Point – Follower Constraints